First of all: with carberry you don't anymore a 5V power supplier. You need to power all system with a +12V power supplier directly from Carberry.Carberry has also a positive ignition input that allows to switch on and off all system by putting Yellow/Blue wire to +12V.

So for the moment:

Connect Red wire to +12V

Connect Black wire to Ground

Connect Yellow/Blue wire to +12V

In this way Carberry will start and will switch on Raspberry pi, after that you will see something on your display. If you will not see Raspbian operating system, try to connect to Raspberry by SSH.Let me know if SSH works or not.

You can connect your raspberry pi to your tv through a wire or wireless HDMI splitter. There are plug and play HDMI switch available in the market. You can easily get one from Amazon. You can check a few reviews best wireless HDMI transmitter and receiver to select the best wireless HDMI before selecting one. Best of luck!!